Abstract

Using some potential theory tools and the Schauder fixed point theorem, we prove the existence and precise global behavior of positive continuous solutions for the competitive fractional system , in a bounded -domain in , subject to some
Dirichlet conditions, where , The potential functions are nonnegative and required to satisfy some adequate hypotheses related to the Kato class of functions .

1. Introduction and Statement of Main Results

Let be a bounded -domain in and be the Dirichlet Laplacian in . The fractional power , , of the negative Dirichlet Laplacian is a very useful object in analysis and partial differential equations; see, for instance, [1, 2]. There is a Markov process corresponding to which can be obtained as follows: we first kill the Brownian motion at , the first exit time of from the domain , and then we subordinate this killed Brownian motion using the -stable subordinator starting at zero. For more description of the process and the development of its potential theory, we refer to [3–6].

In this paper, we will exploit these potential theory tools to study the existence of positive solutions for some nonlinear systems of fractional differential equations. More precisely, we fix two positive continuous functions and on , and we will deal with the existence of positive continuous solutions (in the sense of distributions) for the following competitive fractional system:
where , , and the nonnegative potential functions , are required to satisfy some adequate hypotheses related to the Kato class of functions (see Definition 1). The function is defined by
where is the semigroup corresponding to the killed Brownian motion upon exiting .

We recall that in [6, Remark ], the authors have proved the existence of a constant such that for each ,
where denotes the Euclidian distance from to the boundary of .

In the classical case (i.e., ), there is a large amount of literature dealing with the existence, nonexistence, and qualitative analysis of positive solutions for the problems related to (1); see for example, the papers of Cîrstea and Rădulescu [7], Ghanmi et al. [8], Ghergu and Rădulescu [9], Lair and Wood [10, 11], Mu et al. [12], and references therein. In these works, various existence results of positive bounded solutions or positive blowing up ones (called also large solutions) have been established, and a precise global behavior is given. We note also that several methods have been used to treat these systems such as sub- and supersolutions method, variational method, and topological methods. In [11], the authors studied the system (1) with in the case , , , and , are nonnegative continuous and not necessarily radial. They showed that entire positive bounded solutions exist if and satisfy the following condition:
These results have been extended recently by Alsaedi et al. in [13], in the case , , , , where the authors established the existence of a positive continuous bounded solution for (1).

In this paper, first, we aim at proving the existence and uniqueness of a positive continuous solution (in the sense of distributions) for the following scalar equation:
where and is a nonnegative Borel measurable function in satisfying the following.

The function .

The class of functions , is defined by means of the Green function of as follows.

Definition 1 (see [14]). A Borel measurable function in belongs to the Kato class of functions if

It has been shown in [14], that
For more examples of functions belonging to , we refer to [14]. Note that for the classical case (i.e., ), the class of functions was introduced and studied in [15].

In order to state our existence result, we denote by (see [3]) the unique positive continuous solution of
Using some potential theory tools and an approximating sequence, we establish the following.

Theorem 2. Under hypothesis , the problem (5) has a unique positive continuous solution satisfying for each ,
where the constant .

Using (7), hypothesis is satisfied if verifies the following condition: there exists a constant , such that for each ,
Next, we exploit the result of Theorem 2, to prove the existence of a positive continuous solution to the system (1). To this end, we assume the following hypothesis.

The functions , are two nonnegative Borel measurable functions such that
Then, by using Schauder’s fixed point theorem, we prove the following.

Theorem 3. Under assumption , the problem (1) has a positive continuous solution satisfying for each ,
where and .

We note that contrary to the classical case , in our situation, the solution blows up on the boundary of .

The content of this paper is organized as follows. In Section 2, we collect some properties of functions belonging to the Kato class of functions , which are useful to establish our results. Our main results are proved in Section 3.

As usual, let be the set of nonnegative Borel measurable functions in . We denote by the set of continuous functions in vanishing continuously on . Note that is a Banach space with respect to the uniform norm . When two positive functions and are defined on a set , we write if the two-sided inequality holds on . We define the potential kernel of by
Finally, let us recall some potential theory tools that are needed, and we refer to [14, 16, 17] for more details. For , we define the kernel on by
with , where stands for the expectation with respect to starting from . If satisfies , we have the following resolvent equation:
In particular, if is such that , then we have

2. The Kato Class of Functions

Proposition 4 (see [14]). Letbe a function in, then we have the following.(i). (ii)Letbe a positive excessive function onwith respect to. Then, we haveFurthermore, for each,
we have(iii)The functionis in .

The next two lemmas will play a special role.

Lemma 5. Let be a nonnegative function in and be a positive finite excessive function on with respect to . Then, for all , we have

Proof. Let be a positive finite excessive function on with respect to . Then, by [18, Chapter , proposition ], there exists a sequence of nonnegative measurable functions in such that . Let and such that . Consider , for . Then, by (14), the function is completely monotone on , and so from the Hölder inequality and [19, Theorem ], the function is convex on . This implies that
That is
Hence, it follows from (17) that
Consequently, from (15) we obtain
The result holds by letting .

Lemma 6. Let be a nonnegative function in , then the family of functions
is uniformly bounded and equicontinuous in . Consequently, is relatively compact in .

Proof. Taking in (17), we deduce that for and , we have
So, the family is uniformly bounded.Next, we aim at proving that the family is equicontinuous in .First, we recall the following interesting sharp estimates on , which is proved in [5]:
Let and . By (18), there exists such that
If and , then for , we have
On the other hand, for every and , by using (26) and the fact that , we have
Now since is continuous outside the diagonal and , we deduce by the dominated convergence theorem and Proposition 4 (iii) that
If and , then we have
Now, since as , for , then by the same argument as above, we get
Consequently, by Ascoli’s theorem, we deduce that is relatively compact in .

Lemma 7 (see [14, Proposition 6]). Let and be a nonnegative excessive function on with respect to . Let be a Borel measurable function in such that and . Then, satisfies

Proof of Theorem 2. Let be a positive continuous function on . We recall that on , we have
Let and put , where is given by Proposition 4(i). Since by , , it follows from Proposition 4 that . Define the nonempty closed bounded convex by
Let be the operator defined on by
We claim that maps to itself. Indeed, for each , we have
On the other hand, since the function , we deduce by Lemma 5 with , that . Hence, . Next, we aim at proving that is nondecreasing on . To this end, we let , such that . Using the fact that the function is nondecreasing on , we deduce that
Next, we define the sequence by
Clearly and . Thus, from the monotonicity of , we deduce that
So, the sequence converges to a measurable function . Therefore, by applying the monotone convergence theorem, we obtain
Put . Then, we have
or equivalently
Observe that by Proposition 4(ii), we have . So, applying the operator on both sides of (43), we deduce by using (15) and (16) that
Now, using and similar argument as in the proof of Lemma 6, we prove that . So, is a continuous function in , and is a solution of (5). It remains to prove the uniqueness of such a solution. Let be a continuous solution of (5). Since the function is continuous and positive in such that , it follows that . Then, by using this fact and Lemma 6, we have
So, from the uniqueness of the problem (8) (see [3]), we deduce that
It follows that if and are two continuous solution of (5), then satisfies
where is the nonnegative measurable function defined in by
Since , we deduce by Lemma 7 that , and so .

Proof of Theorem 3. Let and .Put , . Note that from and Proposition 4, we have and . Consider the nonempty closed convex set defined by
Let be the operator defined on by , such that is the unique positive continuous solution of the following problem:
According to Theorem 2, we have
Moreover, we have and and by Lemma 6, is equicontinuous on . Since is also bounded, then we deduce that is relatively compact in . This implies in particular that .Next, we shall prove the continuity of the operator in in the supremum norm. Let be a sequence in which converges uniformly to a function in . Put and . Then, we have
Using the fact that and that , we deduce by Proposition 4 and the dominated convergence theorem, that as . Similarly, we prove that as . So, as . Since is relatively compact in , we deduce that
From the Schauder fixed point theorem there exists such that or equivalently
where . The pair is a required solution of (1) in the sense of distributions. This completes the proof.

Acknowledgment

The research of Imed Bachar is supported by NPST Program of King Saud University; Project no. 11-MAT1716-02.